Of madness and many-valuedness: an investigation into Suszko's Thesis

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Sanderson Molick Silva

Of madness and many-valuedness:

an investigation into Suszko’s Thesis

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UFRN / Biblioteca Central Zila Mamede Catalogação da Publicação na Fonte Silva, Sanderson Molick.

Of madness and many-valuedness: an investigation into Suszko's Thesis / Sanderson Molick Silva. - Natal, RN, 2016.

106 f. : il.

Orientador: Prof. Dr. João Marcos de Almeida.

Dissertação (Mestrado) - Universidade Federal do Rio Grande do Norte. Centro de Ciências Humanas, Letras e Artes. Programa de Pós-Graduação em Filosofia.

1. Tese de Suszko – Dissertação. 2. Bivalência – Dissertação. 3. Consequência lógica – Dissertação. 4. Pluralismo lógico – Dissertação. 5. Redução de Suszko – Dissertação. I. Almeida, João Marcos de. II. Título.

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Of madness and many-valuedness:

an investigation into Suszko’s Thesis

Dissertação de Mestrado apresentada ao Pro-grama de Pós-Graduação em FilosoĄa para obtenção do título de Mestre em FilosoĄa.

João Marcos de Almeida

Orientador

Heinrich Wansing (RUHR -Universität Bochum)

Convidado 1

Luiz Carlos Pinheiro Dias Pereira (UERJ)

Convidado 2

Jean-Yves Béziau (UFRJ)

Suplente

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Acknowledgements

I thank Prof. João Marcos for all the encouragement and support through this whole endeavour. I thank him for being much more than an advisor. I thank him for never accepting less than the best of our efforts.

I thank Prof. Daniel Durante and Prof. David Miller for reading the Ąrst version of this work and making so many important suggestions. I specially thank Prof. Daniel for all the advices given during our brief conversations at the Department halls and for introducing me to logic in the early years of my undergraduation.

I thank Prof. Luiz Carlos Pereira and Prof. Heinrich Wansing for taking part of the defense jury and contributing to the Ąnal version of this thesis. Their reading was crucial in producing the Ąnal outcome. Of course, all remaining mistakes are entirely my fault.

I thank all dear friends who said: ŞVai dar certo, homi!Ť. I thank João Daniel Dantas, Patrick Terrematte, Diego Wendell, Adriano Dodó, Haniel Barbosa, Carolina Blasio, Daniel Skurt, Douglas Cavalheiro, Evelyn Erickson, Hudson Benevides, all the rest of the gang from the Group of Logic at UFRN, and all those with whom I shared my lapses of hope and despair.

I thank CAPES and the GetFun Project for the Ąnancial support.

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Resumo

A Tese de Suszko é uma posição ĄlosóĄca acerca da natureza dos múlti-plos valores-de-verdade. Formulada pelo lógico polonês Roman Suszko, durante a década de 1970, a tese defende a existência de Şapenas dois valores-de-verdadeŤ. Tal aĄrmação diz respeito à concepção de multi-valoração perpetrada pelo lógico Jan čukasiewicz. Considerado um dos criadores das lógicas multi-valoradas, čukasiewicz acrescentou, em adição aos valores fregeanos tradicionais de Verdade e Falsidade, um terceiro valor: o Indeterminado. Para ele, seu terceiro valor poderia ser visto como um passo além da dicotomia Aristotélica entre o ser e o não-ser. De acordo com Suszko, as ideias de čukasiewicz sobre multi-valoração se baseavam em uma confusão entre valores algébricos (aquilo que é descrito/denotado por sentenças) e

valores lógicos (verdade e falsidade). Assim, o terceiro valor-de-verdade criado por čukasiewicz seria apenas um valor algébrico, isto é, uma possível denotação para uma sentença, mas não um valor lógico genuíno. A tese de Suszko encontra respaldo em um resultado formal conhecido hoje como Redução de Suszko, um teorema que aĄrma que toda lógica tarskiana pode ser caracterizada por uma semântica biva-lente. Esta dissertação pretende ser uma investigação da tese de Suszko e de suas implicações. A primeira parte é dedicada às raízes históricas da multi-valoração e introduz as principais motivações de Suszko ao formular a distinção entre valores algébricos e valores lógicos, e assim revelar o caráter duplo dos valores-de-verdade. A segunda parte explora a Redução de Suszko e apresenta seus principais desenvolvi-mentos; as propriedades das semânticas bivalentes em comparação às semânticas multi-valoradas também são exploradas e discutidas. Por Ąm, a terceira parte in-vestiga o conceito de valores lógicos dentro do contexto de noções não-tarskianas de consequência lógica; o signiĄcado da tese de Suszko dentro desses ambientes tam-bém é discutido. Mais ainda, os fundamentos ĄlosóĄcos das noções de consequências não-tarskianas são discutidos à luz do debate recente sobre pluralismo lógico.

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Abstract

SuszkoŠs Thesis is a philosophical claim regarding the nature of many-valuedness. It was formulated by the Polish logician Roman Suszko during the middle 70s and states the existence of Şonly but two truth valuesŤ. The thesis is a reaction against the notion of many-valuedness conceived by Jan čukasiewicz. Reputed as one of the modern founders of many-valued logics, čukasiewicz considered a third undeter-mined value in addition to the traditional Fregean values of Truth and Falsehood. For čukasiewicz, his third value could be seen as a step beyond the Aristotelian dichotomy of Being and non-Being. According to Suszko, čukasiewiczŠs ideas rested on a confusion between algebraic values (what sentences describe/denote) and log-ical values (truth and falsity). Thus, čukasiewiczŠs third undetermined value is no more than an algebraic value, a possible denotation for a sentence, but not a genuine logical value. SuszkoŠs Thesis is endorsed by a formal result baptized as SuszkoŠs Reduction, a theorem that states every Tarskian logic may be characterized by a two-valued semantics. The present study is intended as a thorough investigation of SuszkoŠs thesis and its implications. The Ąrst part is devoted to the historical roots of many-valuedness and introduce SuszkoŠs main motivations in formulating the double character of truth-values by drawing the distinction in between algebraic and logical values. The second part explores SuszkoŠs Reduction and presents the developments achieved from it; the properties of two-valued semantics in comparison to many-valued semantics are also explored and discussed. Last but not least, the third part investigates the notion of logical values in the context of non-Tarskian notions of entailment; the meaning of SuszkoŠs thesis within such frameworks is also discussed. Moreover, the philosophical foundations for non-Tarskian notions of entailment are explored in the light of recent debates concerning logical pluralism.

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The roots of many-valuedness may be traced back to the works of Aristotle and even to the earlier debate between the Eleatic and Ephesian schools of philosophy. Those ancient debates were tied to metaphysical queries about the way we conceive and un-derstand the world, and their inĆuence lasted through the development of Medieval and Modern history of philosophy. However, it was only after the establishment of modern symbolic logic that a formal treatment of the notion of many-valuedness became possi-ble. Thus, from the 1920s, through the works of Jan čukasiewicz, Dmitri Bochvar and Emil Post, many-valued logics were developed and studied from a modern perspective. čukasiewiczŠs motivations were related to AristotleŠs discussions on the law of excluded middle and the problem of future contingents. In order to evaluate propositions about the future, he added a third undetermined value to his logical system and set the theoretical foundations for what is currently recognized as the Ąeld of many-valued logics. Later on, in the middle 70s, the Polish logician Roman Suszko cast doubt on the inĆuential char-acter of čukasiewiczŠs works and the very nature of his scientiĄc enterprise. According to Suszko, the many-valued paradigm of logic was nothing but a ŞhumbugŤ and čukasiewicz was the Şchief perpetrator of a magniĄcent conceptual deceitŤ. For Suszko, truth values play a double semantic role revealed by a difference between what he calls Śalgebraic val-uesŠ and Ślogical valval-uesŠ. The semantic scheme used to express such a dubiety is based on FregeŠs ideas on sense and reference. The referents of sentences, following Suszko, are situ-ations denoted by algebraic values. Even though sentences can describe/denote more than two situations, they are classiĄed by only two logical values: truth and falsehood. This approach led Suszko to claim that Şthere are but two logical valuesŤ, a statement nowa-days recognized as SuszkoŠs Thesis. It Ąnds support in a technical result called SuszkoŠs Reduction, a theorem that shows that every Tarskian logic may be characterized by a bivalent semantics. The present dissertation intends to make a thorough assessment of the issues and developments arising from SuszkoŠs ideas. The philosophical concern about the nature of truth values and the contemporary notions of entailment stand as a pivotal motivation for our work. Furthermore, despite of being aware that some issues adressed in this thesis could be explored in different frameworks, like Ąrst-order or higher-order logics, we shall restrict our attention to propositional logics.

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CONTENTS 15

some important historical elements behind the notion of truth-value since its birth with Gottlob Frege and the classical paradigm of truth-values, until čukasiewiczŠs creation of a third non-classical truth-value. By a classical paradigm of truth-values, we mean the assumption that there are only two truth-values, the True and the False. This thesis is often called as the Principle of Bivalence. Despite other authors had also been responsible for considering non-classical truth-values, we focus only on čukasiewiczŠs motivation for the sake of a better explanation of SuszkoŠs ideas and its philosophical environment. Thus, the chapter begins with a brief presentation of FregeŠs conception of truth-values and some central issues that motivated it. After that, the chapter presents čukasiewiczŠs reasons for considering a logical system with more than two truth-values. At last, SuszkoŠs criticism to čukasiewiczŠs conception of many-valuedness is presented. The main motivations for SuszkoŠs point of view concerning truth-values are exhibited and discussed.

Part II Ű SuszkoŠs Reduction: in the land of bivaluations Ű brings the technical results underlying SuszkoŠs Reduction about the division in between algebraic and logical values. The main result of the chapter of Part II is SuszkoŠs Reduction theorem, which has the purpose of showing that Every Tarskian logic is logically two-valued. Thus revealing logical two-valuedness at the core of the Tarskian notion of entailment. SuszkoŠs Reduction is important for establishing the foundations of bivaluations as an adequate semantic tool in comparison to the matrix theory approach to semantics. However, the move from a matrix semantics to a semantics presented in terms of bivaluations will certainly carry some undesired consequences since some of the structural features inherent to matrices are lost. The chapter then continues by discussing how this consequence may be avoided in light of recent contributions given to SuszkoŠs Reduction.

Part III Ű Beyond SuszkoŠs Reduction Ű discusses generalizations of SuszkoŠs Thesis by presenting alternative notions of entailment constructed from considering additional logical values beyond truth and falsity. The Ąrst chapter begins by introducing Grzegorz MalinowskiŠs ideas about the so-called 𝑞-consequence operations. Malinowski is respon-sible for creating what was baptized by him as inferential many-valuedness. Inferential many-valuedness is the kind of many-valuedness obtained by exploring more than two log-ical values. The addition of a non-classlog-ical loglog-ical value leaves room for the construction of non-Tarskian notions of entailment. After showing how SuszkoŠs Reduction could be generalized to the case of 𝑞-logics, the chapter continues with an exposition of Shramko & WansingŠs conception of logical values and how it affects the usual understanding of SuszkoŠs Reduction. Shramko & WansingŠs construction redeĄnes the usual notion of logic as characterized by a single consequence relation and propose 𝑘-dimensional logics with

𝑘 consequence relations associated to it.

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Part I

On the role of truth values in logical

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1 Truth-values and many-valuedness

Since the development of modern logic and the consolidation of TarskiŠs approach to the notion of logical consequence, truth values played a major role in systems of logic. In fact, the way truth values are deĄned and interpreted according to a given logic is central to deĄne its underlying notion of logical consequence. According to [Shramko and Wansing, 2011], Ştruth values induced a radical rethinking of some central issues in the philosophy of logic and have been put to quite different uses in philosophy and logicŤ. They have been char-acterized as:

∙ Primitive abstract objects denoted by sentences in natural and formal languages,

∙ Abstract entities hypostatized as the equivalence classes of sentences,

∙ What is aimed at in judgements,

∙ Values indicating the degree of truth of sentences,

∙ Entities that can be used to explain the vagueness of concepts,

∙ Values that are preserved in valid inferences,

∙ Values that convey information concerning a given proposition.

Roy Cook [Cook, 2009] highlights that truh-values can be understood as proxies for the various relations that can hold between language and the world. The present chapter intends to give an overview on the birth of non-classical truth values from the works of Jan čukasiewicz. After that, we will discuss the work of some detractors of many-valued logics, paying special attention to the works of Roman Suszko, the Polish logician whose work has inspired a research programme about the meaning and signiĄcance of many-valuedness.

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Scott. Three conception of truth-values underlies the whole development of the chapter, FregeŠs conception of truth-values as abstract entities denoted by sentences, čukasiewiczŠs conception of them as degrees of truth, and SuszkoŠs conception of truth-values as values preserved in valid inferences, as well as the admissible referents of sentences.

1.1 Gottlob Frege, truth-values as reference

Truth-values were called into play by G. Frege [Frege, 1892] as objects denoted by sentences. According to Frege, truth values were mere referential objects denoted as values of arguments to which a concept expression apply. [Frege, 1892] became seminal to the development of contemporary philosophy of language. In it, Frege investigated and developed some concepts not thorougly explained in his Ąrst masterpiece, theBegriffschrift - responsible for launching his famous logicist program. The concepts developed by Frege in those papers were important for formulating his Theory of Meaning for the fragment of language he was concerned since the Begriffschrift.

In the course of investigating the process of grasping the meaning of identity statements, at the core os his account was the distinction between the sense (Sinn) and reference (Bedeutung) of an expresion1

. According to him, whereas the names ŚLewis CarrollŠ and ŚCharles DodgsonŠ may differ in sense, i.e, regarding the cognitive content associated to the expressions, they stand both for the same reference, the actual person corresponding to Charles Dodgson and Lewis Carroll. The sense of an expression is treated as the mode of presentation of its referent, in virtue of which the reference of an expression is denoted.

In FregeŠs theory of meaning, while the reference of a proper name is the object denoted by it, the reference of a complex saturated expression (sentence) is a truth-value, the True or the False. Thus sentences are just names that refer to truth-values. Complex expressions are built using proper names as arguments of functional expressions (concepts). Concept expressions appear in the analysis of predicate expressions such as Śis a cityŠ, Śis bigger thanŠ, Śis a property ofŠ etc. For Frege, all predicates are unsaturated expressions in which the proper names serve as arguments to complete the meaning of the expression. Then, if we have an unsaturated expression like Śis a cityŠ, proper names like ŚNatalŠ, ŚDavidŠ and ŚSeattleŠ serve as arguments making the expression saturated and able to denote a truth-value.

According to [Dummett, 1978], FregeŠs conception of sense and reference lies in the idea that to understand a complete sentence one must think of its truth-value. In

1

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1.1. Gottlob Frege, truth-values as reference 21

this way, the sense of the sentence is a way/procedure to grasp its truth-value. Since the sense of a proper name is a criterion for identifying its referent, the sense of a concept-expression is a way of determining whether or not something satisĄes it. Thus, working these two procedures together, we shall have a procedure for determining the truth-value of a sentence. Moreover, since sense is a mode of presentation of the reference, this implies that truth-values could be understood in a myriad of ways. FregeŠs way out to avoid falling into subjectivism is by placing senses in an objective stance. According to some authors, FregeŠs solution is in accordance with some Neo-Kantian philosophical thesis; we shall explain this in the following.

Some later philosophers, for instance, Strawson [Strawson, 1950] and Davidson [Davidson, 1969] have advocated the strangeness of the idea of treating truth-values as the reference of sentences. However, FregeŠs reasons are connected to the philosophical en-vironment of his age, specially his conception of logic as the Śscience of most general laws of being trueŠ. For Frege, logic is concerned with truth itself, not truth as a mere property of sentences. According to [Gabriel, 1984] and [Gabriel, 2001], FregeŠs philosophical posi-tions were inĆuenced by the Southwest german school of Neo-Kantianism, that emerged under the inĆuence of Hermann Lötze. Moreover, the very use of the word truth-value is connected to the pioneers of that tradition, as highlighted by [Gabriel, 2001], ŞWilhelm Windelband, the founder and the principal representative of the Southwest school of Neo-kantianism, was actually the Ąrst who employed the term Śtruth valueŠ (ŚWarheitswertŠ) in his essay ŞWhat is philosophy?Ť published in 1882 (...).Ť

As founder of the value-theoretical tradition of the Southwest Neo-Kantian school, Windelband deĄned Philosophy as a science of universal values. From his point of view, the main task of Philosophy was to establish universal principles for logical, ethical and aesthetical judgement, thus always oriented by a thelos. Following the way paved by the Windelbandian tradition, Frege opened the paper [Frege, 1956] by deĄning logic in the following manner:

ŞThe word ŚtrueŠ indicates the aim of logic as does ŚbeautifulŠ that of aes-thetics or ŚgoodŠ that of ethics.Ť [Frege, 1956]

Regarding the roots of that tradition, Gabriel [Gabriel, 2001] highlights that the underlying philosophical position of the Southwest school was based in the reunion of a Platonist and Kantian philosophy that emerged from LötzeŠs interpretation of Plato. Such a position was called as transcendental platonism:

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not thought of as an ontological one of existence, but a logical one of being valid.Ť [Gabriel, 2001]

According to [Gabriel, 2001], the position of Frege and some Neo-Kantians (like Windelband) could be described as transcendental platonism. Moreover, the above def-inition could be presented in the form of a transcendental argument which reveals how truth-values appear within the philosophical view purported by the southwest neo-kantian perspective:

ŞLogic starts with making a Śdistinction of valueŠ between Śtruth and un-truthŠ. True and untrue, or false, cannot appear as properties of processes of thinking, but only ofcontentsof thinking. To think about truth and falsehood necessarily pressuposes Ű as a condition sine qua non, that is, as a Ścondition of possibilityŠ in the Kantian sense Ű that we have Ąrst grasped the same cog-nitive content and are discussing the same thought. To take this consequence seriously, we have to accept that a thought cannot be a psychological item, because such a view would imply that different individual subjects are not able to participate in the same cognitive content or thought.Ť [Gabriel, 2001]

The position described above is certainly inĆuential to some later thesis defended by Frege, like his notion of sense as the Gedanke (thought) expressed by a sentence and his rejection of psychologism. However, to what extent Frege might be considered a Platonist or a Kantian is a question that shall not be adressed in this study. The important point to underline is that such independence of thought suggested above implies an item which we want to value as true or false, an item that is meant as the bearer of a truth-value and cannot have individual psychological existence2

. This is important to FregeŠs treatment of the relation between senses and truth-values as reference. However, it is important to remark that despite FregeŠs being inĆuenced by Windelbrand in choosing the wordWarheitswert to refer to truth values, he understands it in a different sense from Windelbrand, because Frege treats truth-values as referents of concept expressions. This is a straight consequence of his mathematical approach to language. How it was explained before, given that concept expressions are predicates which, after being applied to singular terms as arguments, produce sentences, then the values of those functions must be the reference of the sentences.

By considering that the range of functions contain typically objects, then the reference of sentences should be objects, as well 3

. The interesting step taken by Frege here was to treat Śthe TrueŠ and Śthe FalseŠ as objects and not merely as properties. Frege

2

Cf. [Gabriel, 2001].

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1.2. Łukasiewicz and The Possibles 23

understood truth values aslogical objects, that is, mathematical objects such as numbers, sets and alike. He considered truth-values among the grounding objects for his ontology. The major part of FregeŠs later work is dedicated to investigating the nature of such logical objects aiming to achieve a rigorous ontological foundation. In this way, those objects were not only abstract, but also possessed a different ontological import in virtue of their primacy4

.

In the next section, we shall expose Jan čukasiewiczŠs approach to truth-values. He is responsible for deviating from FregeŠs conception of truth-values and being able to postulate the need for a third truth-value, beyond the True and False dichotomy.

1.2 Łukasiewicz and

The Possibles

ŞEntre o sim e o não existe um vão.Ť

Ů Itamar Assumpção

For some authors, a great share of contemporary philosophy emerged from the works of Brentano and his pupils5

. The development and consolidation of Polish philos-ophy was no exception, since the most inĆuential Ągure of the Lvóv-Warsaw School was Kazimierz Twardoswki, one of the three most distinguished students of Brentano (the other two were Alexius Meinong and Edmund Husserl)6

. Naturally, Twardowski was in-Ćuenced by the discussions and subjects explored by his intellectual mentor and dedicated his life to the study of ontology and its relation to language and psychology. To some ex-tent, those were the themes explored by the Ąrst members of the Lvóv-Warsaw school, moved by a scientiĄc conception of philosophy. However, the school inherited not only the philosophical standpoint from the Brentanian tradition, but the Russellian approach to logic and philosophy as well.

The Russellian tradition was introduced by the Ąrst three Polish modern logicians, followers of Russell and Frege: Leon Chwistek, co-inventor of the simple theory of types; Jan čukasiewicz, one of the founders of many-valued logic; and Stanislaw Lesniewski, famous for his great contributions to nominalistic philosophy of mathematics and to con-temporary mereology. Although Meinong is often recognized as one of the founders of the contemporary tradition that seeks to think the nature of non-existent and contradictory objects, according to [Betti, 2011, ], Ş(...) [Twardoswski] was the Ąrst philosopher to hold

4

It is not our concern to adress the question of the adequate meaning of logical objects in FregeŠs work. It is our only goal to describe a little of the history of the concept of truth and how it is related to the modern notion of truth-value.

5

Cf. [Dummett, 2014].

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a theory of intentionality, truth, and predication in which thinking and speaking about non-existents, including contradictions, involves presenting and naming non-existents, in-cluding contradictory objects.Ť As we shall see, the struggle for adequate logical tools for dealing with non-existents, as well as contradictory objects, played an important role in the development of many-valued logics.

Some authors, such as [Rescher, 1968] and [Malinowski, 2009], commonly accredit the Ąrst discussions related to many-valuedness to the ancient Greek philosophers. Re-gardless of that, Rescher [Rescher, 1968] locates the ŞEarly HistoryŤ of many-valued logics within the period from 1875 to 1916. He indicates Hugh MacColl (1837-1909), Charles Peirce (1839-1914) and Nikolai VasilŠev (1880-1940) as the founding fathers of many-valued logic. However, since few formal developments were made by those authors to-wards the creation of a many-valued logical system, Rescher calls ŞThe Pioneering EraŤ of many-valued logics the period from 1920 to 1932, in which Ąrst appeared the works of čukasiewicz and Emil Post. In the present thesis, for the sake of a better explanation of the inĆuences around SuszkoŠs ideas, we shall not talk about Post or any other author from the early history.

1920, the year of the Ąrst publication of [čukasiewicz, 1968], is often mentioned as marking the date of birth of čukasiewiczŠs three-valued logic. Albeit, already in 1918, in his farewell speech at Warsaw University, he asserts:

In 1910 I published a book on the principle of contradiction in AristotleŠs work, in which I strove to demonstrate that that principle is not so self-evident as it is believed to be. Even then I strove to construct non-Aristotelian logic, but in vain. Now I believe I have succeeded in this... I have proved that in addition to true and false propositions there arepossible propositions, to which objective possibility corresponds as a third in addition to being and non-being.(čukasiewicz apud [Wolenski, 1989], p. 119)

As we can notice in the above paragraph, many-valued logic as conceived by čukasiewicz was created from a dissatisfaction of having only truth and falsity as pri-mary notions. The motivations found by him comes from investigations about the nature of science, ontology and probability. Among the main reasons for his abandonment of the classical perspective, are7

:

1) The design of a formal system capable of dealing with the theory of objects proposed by Brentano, Twardowski and Meinong;

2) The problems related to induction and the theory of probability;

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1.2. Łukasiewicz and The Possibles 25

3) The concern with the problem of determinism and its relation to modality.

The Ąrst one has direct connections to the issues dealt with by čukasiewicz in his 1910Šs book: On the principle of contradiction in Aristotle. In it, under the inĆu-ence of MeinongŠs theory about the existinĆu-ence of contradictory objects, objects for which an ontological version of the law of non contradiction fails to hold 8

, čukasiewicz had some intuition towards the necessity of values beyond truth and falsity9

. Furthermore, while contradictory objects, such as MeinongŠs round square, infringe the law of non-contradiction, čukasiewicz also considered abstract objects (like the triangle), called by him Şincomplete objectsŤ, which infringe the law of excluded middle for being free of existence.

As pointed out earlier in this section, čukasiewicz was a product of two tradi-tions, the philosophical approach pursued by those from the Brentanian tradition and the concern with the ontology of logic and its connection to the world, typical from the Frege-Russellian tradition. It is from such a standpoint that it is possible to understand čukasiewiczŠs position regarding the nature of truth values and his creation of the third value. For him, it was clear that the principle of bivalence together with the law of ex-cluded middle made science committed with determinism, because every proposition can only be true or false, specially those about the future since exluded middle ensure the truth of the disjunction despite none of its parts being true10

. For him, statements about the future do not satisfy the excluded middle since they are neither true nor false. As put by [Simons, 1989], čukasiewiczŠs main concern was to make Şscience free from absolute determinismŤ and the way to accomplish such a task should involve the causal necessity that pervades scientiĄc prediction. The strive with future contingents and modality stood out as his main drive in order to formulate an adequate semantics for a three-valued logic. According to [Simons, 2014], čukasiewicz was bothered by the idea of modal logic being trapped into classical bivalent logic. čukasiewiczŠs way out of such problem he added a third value, Śthe possibleŠ, denoted by 1

2. Thus, čukasiewicz believed the signiĄcance of

his three-valued logic was in creating a non-Aristotelian logic.

čukasiewiczŠs system č3 may be deĄned in the following way. Let 𝒱 = ¶0, 1 2,1♢

be the set of truth-values, with 𝒟 = ¶1♢ and 𝒰 = ¶0,1

2♢. The elements of 𝒟 are called

designated truth-values and the elements of 𝒰 are called undesignated. The connectives Ś¬Š, Ś⊃Š and Ś∨Š are deĄned by the following truth-tables:

8

For allaandP: it is not the case that P(a) and¬P(a).

9

Cf. [Simons, 1989].

10

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⊃ 0 1 2 1

0 1 1 1

1 2

1

2 1 1

1 0 1

2 1 ¬ 0 1 1 2 1 2 1 0

∨ 0 1 2 1

0 0 1

2 1 1 2 1 2 1 2 1

1 1 1 1

We say a formula is a tautology if it is always assigned a designated value. In this case, note that,Ð∨ ¬Ðand ¬(Ð∧ ¬Ð) are not tautologies in č3. Later on, čukasiewic also

exhibited how to extend his system in having Ąnitely or inĄnitely many truth-values. For him, truth-values within those systems expressed the truth degree of sentences. The idea set the basis for what is nowadays called Fuzzy logics.

It is important to mention that čukasiewiczŠs conception of truth-value was based on the technical tools explored by the polish logicians, what also moved a mathemati-cal conception of truth-values. According to [Béziau, 2012], thismathematical conception of truth-values appears in its complete form in the history of logic only after the devel-opment of the notion of logical matrices11

. In [Béziau, 2012], the author describes the mathematical concept of truth value in the following way:

ŞLet us have a look at the MTV [Mathematical concept of Truth Value]. It is a double structure: on the one hand we have an absolutely free algebra, on the other hand, facing it, an algebra of similar type, Ąnite or not, and between them, the central notion establishing relations between mathemati-cal structures, the notion of morphism. The elements of the free algebra are called propositions and its functions connectives, the elements of the facing structure are called truth-values and its functions truth-functions, and Ąnally the morphisms between the two structures valuations.[Béziau, 2012]Ť

From this perspective, truth-values are only mathematical elements from an al-gebra of an adequate similarity type. They are divided by distinguishing proper subsets of them into designated and non-designated. From this, each morphisms between the structures plays the role of assigning the propositions that receive a designated value and, therefore, have a model, from the ones which does not have a model. The mathematical ap-proach to the concept of truth-value led us to deĄne properties such as truth-functionality, analyticity, and so on, in a proper manner. The development of such mathematical struc-ture of truth values owes a great debt to Boole and Peirce. However, it was Tarski and Lindenbaum the Ąrst ones to set the algebra of formulas and truth values in a clear ab-stract perspective. Their treatment established the foundations for the total abab-straction of the concept of logic by looking at the structural features of the notion of entailment.

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1.3. Roman Suszko on algebraic and logical values 27

Therefore, the connections between truth values and logical consequence were thus prop-erly revealed and treated. In this regard, as we shall discuss in Part III of the present thesis, the notion of logical consequence became prominent in deĄning what a logical system is.

1.3 Roman Suszko on

algebraic

and

logical

values

Roman Suszko (1919-1979) is one of the most prominent modern logicians from the Lvóv-Warsaw school. He is accredited with an extensive work on abstract logic, on model theory, and on the algebraic tradition initiated with Alfred Tarski. Suszko appeared in PolandŠs philosophical scenario as a contemporary of čukasiewicz, therefore during the mix between the Brentanian and the Russellian tradition. Despite the contact of the Brentanian tradition on the philosophical approach pursued by the Lvóv-Warsaw school, the main inĆuences for Suszko, on what regards his treatment of logic and ontology, are the Ągures of Frege and Wittgenstein.

SuszkoŠs concern with the nature of truth values became central in his work only by the end of his career. The famous paper in which he develops his thoughts on truth values is [Suszko, 1975a], in which he addresses the discussion about the expressive power of so-called non-Fregean logics (NFL), logics characterized by the failure of the so-called Fregean Axiom (FA). The paper is a presentation and development of the most sim-pliĄed version of NFL, the so-called Sentential Calculus with Identity, (SCI), that is a product of other investigations by Suszko and his co-authors like [Bloom et al., 1972] and [Bloom and Suszko, 1971]. For Suszko, NFL was responsible for continuing FregeŠs pro-gram without the Fregean Axiom, what could be seen as equivalent to Şrealising EuclidŠs program without the Ąfth postulateŤ.

SuszkoŠs conception of truth values and the ontology of logic provided the the-oretical motivations for his creation of NFL. His ideas were elaborated in the context of struggling with the Fregean axiom and its implications. He used the expression non-Fregean logics to refer to the class of logics that do not satisfy the following principle:

(FA1) all true (and, similarly, all false) sentences describe the same, that is, have a common referent.

(FA1) is recognized as the general version of the Fregean Axiom12

. Following FregeŠs ideas, since the referent of sentences are truth values and there are only two of them, then all true sentences denote the truth value True, whilst all false sentences denote the truth value False.

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Suszko starts the paper [Suszko, 1975a] by interpreting FregeŠs semantic scheme in the following way: given a sentence ã, we shall call 𝑟(ã) the referent of the sentence,

𝑠(ã) the sense ofã, and 𝑡(ã) its logical value. In FregeŠs theoretical construction, once the reference of saturated sentences are truth values, we have the following conditions on the assignments, given sentencesã and å:

𝑟(ã)̸=𝑟(å) implies 𝑠(ã)̸=𝑠(å) (1.1)

𝑡(ã)̸=𝑡(å) implies 𝑟(ã)̸=𝑟(å) (1.2)

The converse of (2), according to Suszko, is one version of the Fregean Axiom, since it tell us that different referents must receive different logical values. The whole point of SuszkoŠs construction with NFL is in negating such idea by allowing that different referents may receive the same logical values. For him, the way to accomplish depends on being able to draw a difference between logical values and the referents of sentences.

By still making use of FregeŠs terminology of sense and reference, Suszko deĄned the referent of a sentenceãas the situationdenoted by ãand the sense as theproposition expressed by ã. Based on this, we can mention two reasons for SuszkoŠs rejection of the Fregean Axiom: 1) The fact that equating reference and logical values implies the existence of only two possible situations described by sentences, and 2) It entails the confusion between what sentences describes/denote and their logical values. Therefore, by evading the Fregean Axiom, one would obtain a richer ontology with more than two possible referents.

In order to avoid the problem of having at most two possible situations as reference, caused by the Fregean Axiom, Suszko enriched the language of propositional logic with a new operator Ś⊕Š, which is used to assert the identity of situations, i.e, the identity of what the sentences describe. By taking classical propositional logic plus Ś⊕Š, we obtain SCI if we impose the following constraints upon Ś⊕Š 13

:

𝜙 ⊕𝜙 (1.3)

(𝜙⊕å)⇒à[𝑝↦⊃𝜙]⊕à[𝑝↦⊃å] (FA3)

(𝜙 ⊕å)⇒(𝜙 ⊃å) (1.4)

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where à[𝑝 ↦⊃ Ð] denotes the substitution of every 𝑝 in à by Ð. The remaining symbols stands as usual.

The connective Ş⊕Ť may also be introduced in the following way:

𝑡(ã⊕å) = 1 iff 𝑟(ã) =𝑟(å) (1.5)

Note that, (1.5) states thatã⊕å takes the value 1 if and only ifã andå describe the same situation. One of SuszkoŠs goals in deĄning Ś⊕Š in the above manner is in avoiding what he calls the ontological version of the Fregean Axiom, namely the converse to 1.4:14

.

(ã ≺å)⇒(ã ⊕å) (FA2)

For Suszko, the problem with (FA2) is in confusing identity of situations and material equivalence. In [Suszko, 1975a], he presented an axiomatic basis for NFL, called 𝒲 (for Wittgenstein), and shows how to construct stronger systems by adding different axioms to 𝒲 (what gives us the hierarchy of kind 𝒲 systems). The strongest system of the hierarchy, called 𝒲ℱ, is obtained from 𝒲 by adding the ontological version of the Fregean Axiom. Moreover, according to Suszko, the different systems between 𝒲 and 𝒲ℱ, constructed by imposing different constraints, stands for different Şontological principles concerning the structure of the universe of situationsŤ. That is why he believed that (FA) represents a strong ontological constraint, because, as was explained above, it limits the set of possible situations to at most two.

Another interesting point of kind𝒲 systems is in exploring the relation between material equivalence and identity. This relation is a theme well explored in Suszko & BloomŠs sentential calculus with identity (SCI). According to them, SCI is the Şweakest extensional two-valued logicŤ. SuszkoŠs obsession with an adequate treatment for identity and material equivalence relies on his appreciation of extensionality and his fear of Şinten-sional ghostsŤ. Therefore, in [Suszko, 1975a], he discusses the truth-functional character of identity and even claim that Ş[identity] is truth functional and, then, coincides with material equivalence, if and only if the Fregean axiom holdsŤ.

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On what regards SuszkoŠs conception of reference, it is based on the notion ofstates of affairs, exposed by Wittgenstein in the Tractatus. In [Suszko et al., 1968], he elaborated a system intended as a formalized version of the ontology developed in the Tractatus. (He points out the monograph written by the logician Boguslaw Wolniewicz [Wolniewicz, 1968] as his primary source of inspiration.). By following the Tractarian perspective on reference, Suszko is avoiding FregeŠs neo-Kantian perspective on truth values. Therefore, Suszko is building the path toward an alternative conception about reference and the role of truth values as semantic entities. According to Malinowski [Malinowski, 2009], ŞOne could say that SuszkoŠs interpretation of truth-values rests on the distinction between two semantic levels: ontological and logicalŤ. Thus, SuszkoŠs account tries to separate the ontological world standing as the denotation of sentences from the logical notions of truth and falsity.

According to SuszkoŠs later papers, [Suszko, 1975a] and [Suszko, 1977], 𝒱, the set of truth-values, stands as the set of algebraic values and each of its elements denotes the possible referents of sentences, i.e., situations. Moreover, the distinction of the alge-braic values into two subsets𝒟 and 𝒰, called, respectively,designated and undesignated, represents the two genuine logical values, the adequate notions of truth and falsity.

The Ąrst paper written by Suszko about the concept of logical values dates from 1957, entitledFormal theory of logical values; unfortunately, it never received an English translation 15

. An important consideration made by Suszko regarding the concept of logical value appeared as a deĄnition of logical value in an encyclopedia called Notions and theorems of elementary formal logic [Pogorzelski and Pogorzelski, 1994]:

ŞGenerally speaking, the term logical value in a metalanguage of a certain propositional logic refers to every element of characteristic matrix of that logic or (more generally) an arbitrary element of a universum of an arbitrary logical matrix of the language 𝑆0. In the early stages of the development of

modern logic, truth and falseness (which were referred only to propositions or propositional expressions) were called logical values. When it was assumed that a sentenceÐ was true, then it was said that it had a logical value of truth and it was possible to replace it by the symbol 1 or𝑇; when it was false, it was possible to be replaced by the symbol 0 or the sign 𝐹 (0, 1 Ů symbols taken from Boolean algebra). However, this point of view was not uniform: as logical values were treated both as truth and falseness (intuitively understood), and the symbols 0, 1, or as symbols of a Ąxed but arbitrary Ů false or true sentence. In the last case, it was a linguistic understanding of logical values. Ť Suszko: Formal theory of logical values apud [Pogorzelski and Pogorzelski, 1994]

In the above passage Suszko claims the term logical value refers to every element

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of a matrix, what is contradictory to our separation between algebraic and logical values. However, in the same passage, Suszko goes on and expresses dissatisfaction with such use of the term logical value and how it has been treated since the birth of so-called many-valued logics:

ŞGood foundation of such a formulation of the notion of logical value was PostŠs proof showing that classical propositional logic in axiomatic form has two-valued characteristic matrix. However, almost at the same time logics with characteristic matrices of numbers bigger than two occurred Ů e.g., čukasiewiczŠs three-valued logic (the term logic is understood here as set of tautologies of a matrix). This third element of universum of this matrix was traditionally interpreted as possibility.

Logical values have quickly lost their philosophically intuitive interpreta-tions when logics, whose characteristic matrices were𝑛-ary, occurred (with an arbitrary, natural 𝑛) or could even be of inĄnite cardinality.

Obviously, logical values (many-valued logics, in contradistinction to two-valued logic) were Ů and are Ů discussed but these logical values have lost their intuitive sense, although an opinion was stated that they could be degress of truthfulness of propositions.

(...)

As the propositional logics and their metatheories were developing, the va-riety of characteristic models (matrices) and elements of their carriers (from rational numbers to topological spaces) increased and, according to that, it is stated that by logical values are understood elements of a universum of logical matrix, which because of intuitive or philosophical reasons, are iden-tiĄed with the notion of truth, falseness or their variants (necessity, possi-bility, randomness). However, when such identiĄcations are not made and intentions of founders of a certain logic do not lead to a simple split of the notion of truth or falseness and so making a logic with many logical values instead of two, then a certain logic is not called many-valued and elements of universum of characteristic matrix are not always called logical values. Thus, this notion is not described properly and the fact whether a certain element of characteristic matrix of a certain logic is called logical value or not depends on non-formal means, intuitions or a predilection of a founder of a logic or its main users. Ť Suszko: Formal theory of logical values apud [Pogorzelski and Pogorzelski, 1994]

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value. Suszko is interested on the way we understand truth values within the mathemati-cal structures we use in mathematimathemati-cal logic. His remark about the good foundation found in PostŠs result had some inĆuence on his project of showing that every many-valued semantics can be characterized by a bivalent one, a result nowadays called Suszko’s Re-duction. Even though Suszko [Suszko, 1975b] and Malinowski [Malinowski, 1990a] have made some contribution to the way SuszkoŠs Reduction may effectively be carried out, an algorithmic procedure for such a task was exhibited only later on by [Caleiro et al., 2007].

Given the distinction between algebraic and logical values, we see that truth values play a dubious semantic role in a logical system. This ambiguity reveals itself by highlight-ing the fact that, in SuszkoŠs terminology, the expression truth values denotes different things, e.g, the elements inside the matrix, as well as its partitions. From this, hereafter we shall adopt a terminology to refer to the different modes in which many-valuedness may express itself by using the expressions referential and inferential many-valuedness. The former is about the cardinality of the designated set of truth values, whereas the latter concerns the number of partitions of the matrix 16

. This difference will be important in order to stress the meaning of SuszkoŠs Thesis and its later developments.

From what was said above, we can conclude that there were two central reasons for SuszkoŠs criticism of the Fregean Axiom: 1) the fact that it collapses algebraic and logical values, 2) that it entails the confusion between identity and material equivalence. Only the former shall be thorougly explored in thein present study in virtue of its direct connection with the formulation of SuszkoŠs Thesis and how it was understood and developed in the literature 17

.

1.4 Suszko’s Thesis and many-valuedness

During the 1970s, in a talk delivered at the 22nd _{Conference on History of Logic}

(Craców), Roman Suszko called into question the nature of the logical enterprise pursued by Polish logic since the 1920s under the inĆuence of čukasiewicz [Suszko, 1977]. Suszko striked and accused čukasiewicz to be Şthe chief perpetrator of a magniĄcent concep-tual deceit lasting out in mathematical logicŤ and provoked: Şhow was it possible that the humbug of many logical values persisted over the last Ąfty years?Ť. SuszkoŠs ideas questioned the notion of many-valuedness proposed by čukasiewicz.

Based on the above difference between algebraic and logical values, Suszko asked ŞHow could he [čukasiewicz] confuse truth and falsity with what sentences describe?Ť, the difference that led him to claim Şthere are but two genuine logical valuesŤ. His ideas

16

The expressions referential and inferential many-valuedness were not used by Suszko. Instead, the Ąrst authors to use it were Wójcicki and Malinowski.

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1.4. Suszko’s Thesis and many-valuedness 33

endorse the fact that, while the classical conception of truth values might be denied at the referential level, i.e., at the level of what sentences describe, it remains true at the inferential level, through the Tarskian notion of inference. The reason for this is in the fact that TarskiŠs notion of entailment depends only on the dichotomy between the designated and undesignated sets of truth values. Then, once for Suszko the genuine notions of truth and falsity are expressed by logical values, they are our genuine values and it would be a Şmad ideaŤ to have more than two of them. This philosophical position concerning the nature of many-valuedness became known and referred to in the literature as Suszko’s Thesis.

In [Suszko, 1975b], Suszko exhibited a sketch of a bivalent description of čukasiewicz three-valued logic and suggested that such approach was strong enough to be applied to any many-valued logic. However, the Ąrst one to give a Ąrst step in showing that SuszkoŠs approach could be applied to any many-valued logic was Malinowski in [Malinowski, 1990a]. Nevertheless, the Ąrst ones to show a full description on the procedure of Ąnding an ade-quate bivalent semantics to any many-valued semantics were Caleiro et al [Caleiro et al., 2003]18

.

As was said above, Frege assumed that there were only two referents of sentences: the True and the False. All true sentences denote the True, and all false sentences denote the False. By adopting a Tractarian perspective on reference, Suszko takes a realist stance and builds a richer ontology of situations. Moreover, his division of algebraic and logical values had the importance of defending the idea that sentences assigned the same logical value need not denote the same.

For Suszko, as for Wittgenstein, the world is conceived as the totality of facts (situations). The classiĄcation into designated and undesignated values has the purpose of selecting the situations that obtain and the ones that do not. This idea is obviously very close to WittgensteinŠs Tractarian construction in dividing the world into negative and positivefacts19

. According to Suszko, an adequate formalized language to deal with objects and situations must have two types of variables: nominal variables running through the universe of objects and sentential variables running through the universe of situations. This construction set the basis for NFL.

The importance of such concept of reference is in going against the Fregean con-ception of truth values. Suszko did not take logic as a machinery tool to discover truths about an ontological realm beyond human direct experience. Instead, Suszko believes that formal languages are created from the attempt to grasp some fragments of reality. Accord-ing to Omyla [Omyla,], for Suszko Şthe subject-matter of logical investigations are any conceptual structures emerging from the process of world cognitionŤ. Moreover, Şthere is

18

Here we take a bivalent semantics as any arbitrary family of functions from the set of propositional formulas to the set𝒱2=¶0,1♢. Font [Font, 2009] called such notion of semanticsSusko’s semantics. 19

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a structural syntatic framework, by means of which consciousness can grasp reality.Ť Thus the logical structure of a language related to the fragment of reality it tries to formalize is never arbitrary and purely linguistic but is determined by20

:

(a) the ontological structure of the fragment of reality to which the language refers. (b) the semantic principles adopted.

SuszkoŠs position is similar to the Wittgensteinean conception of the interaction between reality and the logical structure of language, by which formal languages are able to exhibig the logical form of reality. He takes logic as dealing with some aspects of the logical structure of the world, what is supported by his view on the ontology of the world as the totality of situations. Thus, his realist conception of reference serve as foundation for a relational general theory of reference, where situations are seen as blocks of reality that logical systems tries to grasp. Although he had not entirely developed a theory of situations21

, Suszko was aware of the importance of taking situations as primary ontological entities. In [Suszko, 1994], he took events as objects abstracted from situations and proved that some theories of situations are mutually translatable into theories of events.

SuszkoŠs concern is directed toward drawing a sharp distinction between the levels in which many-valuedness may be expressed by establishing the difference of algebraic and logical values. The notions of algebraic and logical values are useful to relate a rich ontology of situations as referents, and yet avoid clouded concepts related to undetermined truth or falsity. By doing that, Suszko gives support to fundamental notions of truth and falsity and helps to clarify the nature of many-valuedness in logic.

1.5 Michael Dummett and Dana Scott on many-valuedness

Despite the richness of SuszkoŠs thoughts, he was not the Ąrst one to contend against many-valuedness. Other authors such as Michael Dummett ([Dummett, 1991] and [Dummett, 1978]) and Dana Scott ([Scott, 1973] and [Scott, 1974]) made, independently, similar criticism regarding the way many-valuedness was treated and deĄned. In what follows, we present these criticisms and how they relate to each other.

The precise sense in which we claim that DummettŠs and SuszkoŠs comments are similar lies on the fact that both of them considered truth and falsiy as concepts expressed by the Tarskian dichotomy of designated and undesignated values. Already in 1959, in [Dummett, 1978] and also [Dummett, 1991], Dummett analyzes the different uses of truth and falsity as performed by some utterances. However, differently from Suszko, Dummett

20

Cf. [Omyla,].

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1.5. Michael Dummett and Dana Scott on many-valuedness 35

is not worried with an ontological principle behind the treatment of truth values, but in setting a in treating the different uses of truth and falsity.

In the preface of [Dummett, 1978], the author says:

ŞTruth(...) was a defense of the principle of tertium non datur22

against certain kinds of counterexample; not, of course, that I wanted to contend against uses of ŚtrueŠ and ŚfalseŠ under which an utterance could be said to be recognised, in certain senses, as being neither true nor false, so long as the point of using those words in such a way was acknowledged to be only to attain a smoother description of the way the sentential operators worked.Ť

Thus, according to Dummett, we can accept different or intermediate notions of truth and falsity as long as they are used only to get a better description of the connectives. This point is very similar to SuszkoŠs later consideration on algebraic valuations, which are characterized by a homomorphism between the algebra of formulas and the algebra of truth values. The bivalent characterization of the algebra of truth values, of course, would lose its homomorphic structure, thus not possessing a ŞsmoothŤ description of the connectives. In [Dummett, 1978], in the context of discussing a logic with 𝑇, 𝐹, 𝑋 and

𝑌 as truth-values, Dummett remarks:

ŞLogicians who study many-valued logics have a term which can be em-ployed here: they would say𝑇 and 𝑋 are ŚdesignatedŠ truth-values and𝐹 and

𝑌 ŚundesignatedŠ ones. (In a many-valued logic those formulas are considered valid which have a designated value for every assignment of values to their sentence-letters). The point to observe are just these. (i) The sense of a sen-tence is determined wholly by knowing the case in which it has a designated value and the cases in wich it has an undesignated one. (ii) Finer distinctions between different designated values or different undesignated ones, however naturally they come to us, are justiĄed only if they are needed in order to give a truth-functional account of the formation of complex statements by means of operators. (iii) In most philosophical discussions of truth and fal-sity, what we really have in mind is the distinction between a designated and an undesignated value, and hence choosing the names ŚtruthŠ and ŚfalsityŠ for particular designated and undesignated values respectively will only obscure the issue.(...)Ť [Dummett, 1978]

Dummett then reveal similar ideas to SuszkoŠs distinction on algebraic and logical values. For Dummett, though we can have statements that may seem neither true nor

22

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false regarding its content, they are all classiĄed as designated or undesignated. There-fore the genuine conceptions of truth and falsity are expressed in the distinction des-ignated/undesignated. In chapter 2 of [Dummett, 1991], Dummett draws a distinction between ingredient sense and assertoric content of sentences. A distinction very similar to the difference of algebraic and logical values. DummettŠs approach, however, is justiĄed by the content of some utterances, which may be regarded as neither true nor false and, at the formal level, they are described by undetermined truth values only to attain an adequate functional description of the operators. In [Dummett, 1978], the author gives the following example:

ŞI once imagined a case in which a language contained a negation operator Ś`Š which functioned much like our negation save that it made Ś`(𝐴 ⊃ 𝐵)Š equivalent to Ś𝐴⊃_` 𝐵Š, where ⊃ is the ordinary two-valued implication. In this case, the truth or falsity of Ś`(𝐴 ⊃𝐵)Š would not depend solely on the truth or falsity of Ś𝐴⊃𝐵Š, but on the particular way in which Ś𝐴⊃ 𝐵Š was true (whether by the truth of both constituents or by the falsity of the an-tecedent). This would involve the use of three-valued truth tables, distinguish-ing two kinds of truth. In the same way, it might be necessary to distdistinguish-inguish two kinds of falsity.Ť

L. Humberstone [Humberstone, 1998], explains that the difference between asser-toric content and ingredient sense is introduced Şin terms of the distinction between knowing the meaning of a statement in the sense of grasping the content of an assertion of it and in the sense of knowing the contribution it makes to determining the content of a complex statement in which it is a constituent.Ť Sentences that are alike regarding the assertoric content (have the same designated value) are true under the same conditions. Therefore, despite sentences being equal regarding their assertoric content (in SuszkoŠs terminology, the logical value), they differ relative to their ingrediente sense (the algebraic value).

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1.5. Michael Dummett and Dana Scott on many-valuedness 37

¶t,f♢-valuations, but few Ů even the creators of the subject Ů can understand many-valued truth tablesŤ.

Aiming to discuss the notion of inference related to čukasiewicz logic, Scott presents his criticism to many-valued logics in the following way23

. Take, for instance, čukasiewiczŠs 3 valued conjunction with values 𝒱 =¶0,1,2♢,𝒟 =¶1♢ deĄned in the following way :

∧ 0 1 2

0 0 0 0

1 0 1 1

2 0 1 2

where 2 corresponds totrue and 0 tofalse. However, in order to avoid calling themvalues, Scott advises to treat them as types of sentences. In this way, we could set 𝑆 =¶0,1,2♢ and map the set of propositions to their respective types. Moreover, he considers that čukasiewiczŠs intuitions about designation are vague and stresses ŞIs not the division of statements types into the designated and undesignated just a truth-valuation? Of course. So why not call it one?Ť In this way, there are many valuations that we could consider. We can show them in a table:

𝑉 𝑣0 𝑣1

0 f f

1 f t

2 t t

where𝑉 =¶𝑣0, 𝑣1♢, consists of two valuations able to distinguish three types of sentences.

Each valuation therefore represents one way of designating the elements. From this, we can deĄne an entailment relation in the following way:

ΓÐ iff Γ♣=V Ð, for all 𝑣 ∈𝑉. (1.6)

and it is straightforward to see that Ð iff 𝑣0(Ð) =t. For instance, the truth-conditions

for conjunction can be described by using the valuations as:

𝑣i(Ð∧Ñ) = tiff 𝑣i(Ð) = tand 𝑣i(Ñ) =t (1.7)

𝑣i(Ð∧Ñ) = f iff 𝑣i(Ð) = f or𝑣i(Ñ) =f (1.8)

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where 0 ⊘ 𝑖 < 2. Here the numbers work only as the subscripts for the valuations. After showing this kind of construction, Scott then considers, Şit seems much better to consider a variety ofvaluations rather than a variety of ŚtruthŠ values. Valuations use the ordinary truth-values,tandf, and sentential ŚvaluesŠ could creep in again (...) as types of sentences as distinguished by the valuations (...)Ť. ScottŠs construction makes it possible to obtain a bivalent characterization of the original many-valued semantics24

. This sets the foundations for his criticism to many-valued logics. In the next chapter, we use a similar approach to prove SuszkoŠs reduction theorem, a result that establishes and reveals the bivalent character of many-valued Tarskian systems.

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Part II

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41

2 Exploring Suszko’s reduction

Before starting to list the main points of this chapter, some remarks about the way we shall deĄne logic and the understanding that shall drive the present study might be necessary. Such a conception of logic (see Definition 5 below) is borrowed from Jean-Yves Béziau from what he baptized as Universal Logic. The history of Universal Logic, understood here as a general theory of logics and not as a universal system of logic (as some might believe) begins with Alfred Tarski. Tarski launched his theory of an abstract consequence operator with the intention of describing the processs of reasoning underlying the methodology of deductive sciences. Despite the inĆuential character of TarskiŠs ideas on the abstract theory of logics, like the one pursued by Brown, Suszko and Bloom [Brown et al., 1973], some other abstract approaches were developed independently, like the idea of sequents by Hertz and Gentzen 1

. The importance of Tarski and the others lies in the fact that they created tools to investigate the notion of deduction apart from the traditional Hilbert-style proof systems 2

. Along with that development, several types of semantic tools, such as model theory and matrix theory, were developed to characterize the semantic notion of consequence subjacent to logical systems. This chapter is about how bivaluations are related to matrix theory as an adequate semantic tool to characterize TarskiŠs notion of consequence.

Bivaluations are not new animals in the logical zoo. We might say that they abound in the logical realm since its modern foundations and they appeared as a semantic tool with the creation of classical logic and its semantics. The aim of this chapter is for it to be an investigation about the range of applicability of bivaluations as an appropriate semantic tool for logical systems by analyzing their fundamental properties in comparison to the usual matrix semantics. The Ąrst section is a detailed exposition of the central result that led Suszko to formulate his ideas about algebraic and logical many-valuedness. The result was important in establishing the foundations for studies that sought to develop the theory of bivaluations as a general machinery to construct semantics for different logical systems. On that regard, an important step was taken by Newton da Costa and collaborators in [Loparic and da Costa, 1984], [da Costa and Béziau, 1994] in proposing what is known as the Theory of Valuations. Newton da CostaŠs theory of valuations was intended as an abstract bivalent framework for developing semantics for arbitrary logical systems. The second section aims to be an exposition of the major contributions made to the bivalent reduction procedure initiated by Suszko. Based on Caleiro & MarcosŠs reduction procedure, we will illustrate how to obtain a bivalent semantic characterization

1

Cf. [Beziau, 2005].

2

(44)

of GödelŠs 3-valued logic. We also point out how the procedure can be applied to any Ąnite-valued logic.

2.1 On the meaning of Suszko’s reduction

This section introduces some of the main results that laid the foundations for SuszkoŠs Thesis, as well as the major developments based on SuszkoŠs ideas. We begin by presenting the technical machinery that will be useful to prove SuszkoŠs fundamental result, the well-knownSuszko’s Reduction theorem Ű that states that every Tarskian logic may be characterized by a bivalent semantics. The theorem is responsible for showing that many-valued semantics may be recognized as bivalent semantics in disguise. After-wards, we will present a sketch of how the technique for SuszkoŠs reduction procedure was improved by Carlos Caleiro & João Marcos in order to extend its range of application and provide a recipe for carrying it out.

Fundamental concepts

We begin by introducing some basic terminology regarding the concept of an ab-stract algebra.

Definition 1. An algebraic type is a pair á =⟨𝐹, 𝜌⟩ where 𝐹 ̸= ∅ is a set of symbols and 𝜌:𝐹 ⊃N _{is the arity function, the map that assigns an arity to each symbol in} _𝐹_.

Definition 2. Analgebraof typeá is a structureA₌_⟨𝒜_,_𝒪⟩_where_{𝒜 ̸=}_∅_{is the domain} of the algebra (the carrier set) and 𝒪=¶𝑓A

i ♢i∈F such that for all 𝑖∈𝐹:

if 𝜌(𝑖) = 𝑛, then 𝑓A

i :𝒜

n _{⊃ 𝒜} _(2.1)

Given two algebras A_and B_{, we say they have the same type in case} _á₍A_{) =} _á₍B_).

Definition 3. The notion of homomorphism between algebras is defined in the follow-ing way. Let A _and B _{be two algebras of the same type. We say that} _ℎ _: _{𝒜 ⊃ ℬ} _{is a} homomorphism of A _into B _{if and only if for every} _𝑛_{-ary function} _𝑓A

i , we have that

ℎ(𝑓A

i (𝑎1, ..., 𝑎n)) =𝑓

B

i (ℎ(𝑎1), ..., ℎ(𝑎n)) (2.2)

The set of all homomorphisms of A_into B _{is denoted by} _𝐻𝑜𝑚₍A_,B_{). If a homomorphism} is from a given algebra A _{into itself} ₍_ℎ _:_{𝒜 ⊃ 𝒜), then it is called an} _endomorphism_. The set of all endomorphisms on A _{is denoted by} _𝐸𝑛𝑑₍A_).

Now we introduce our algebra of formulas, Ąrst let𝐴𝑡=¶𝑝1, 𝑝2, ...♢be a

denumer-able set of atoms, and let Σ =¶Σn♢n∈N be a propositional signature, where each element

Of madness and many-valuedness: an investigation into Suszko's Thesis

Sanderson Molick Silva

Of madness and many-valuedness:

an investigation into Suszko’s Thesis

Of madness and many-valuedness:

an investigation into Suszko’s Thesis

Acknowledgements

Resumo

Abstract

Contents

I On the role of truth values in logical consequence

17

II Suszko’s reduction: in the land of bivaluations

39

III Beyond Suszko’s Reduction

65

Introduction

Part I

On the role of truth values in logical

1 Truth-values and many-valuedness

1.1

Gottlob Frege, truth-values as reference

1.2

Łukasiewicz and

The Possibles

1.3

Roman Suszko on

algebraic

and

logical

values

1.4

Suszko’s Thesis and many-valuedness

1.5

Michael Dummett and Dana Scott on many-valuedness

Part II

2 Exploring Suszko’s reduction

2.1

On the meaning of Suszko’s reduction

Fundamental concepts